NASA Technical Memorandum 105326

/A/_JO

Development of a Single-Phase HarmonicPower Flow Program to Study the 20 kHz

A.C. Power System for Large Spacecraft

L. Alan Kraft

Valparaiso University

Valparaiso, Indiana

and

M. David Kankam

Lewis Research Center

Cleveland, Ohio

November 1991

:_9Z- 13275

https://ntrs.nasa.gov/search.jsp?R=19920004057 2020-06-19T09:18:48+00:00Z

TABLE OF CONTENTS

NOMENCLATURE ..................................................... iii

Abstract .............................................................. 1

1. Introduction ......................................................... 1

2. Operation of the Mapham Inverter ..................................... 3

3. Modelling the MaDham Inverter ....................................... 4

4. Operation of the Single Phase. Voltage Controlled Rectifier. ............. 11

5. Modelling the Single Phase. Voltage Controlled'Rectifier. ............... 12

6. The Harmonic Power Flow Algorithm .................................. 17

7. Modifying the Harmonic Power Flow Algorithm ......................... 21

8. _ ........................................................... 24

9. Conclusions and Recommendations .................................. 27

LIST OF REFERENCES ................................................ 29

Appendix A ........................................................... 30

PRECEDING PAGE BLAP.]K P,_O_!FILI'_EL)

iii

NOMENCLATURE

a

AC

[A]

[A]r

[Ao_]

A (i)

AA

(Z

b

C

C

d

D

D'

DC

f

GDC

I

I(s)

im[A]

J

[J]

[Jij]

L

Coefficient of a polynomial

Alternating current

Matrix A

Transpose of matrix A

Matrix A at the im iteration

The i th harmonic component of A

The change in A

Real component of a complex frequency

Coefficient of a polynomial

Real component of a complex Fourier coeffiecient

Capacitance

Imaginary component of a complex Fourier coefficient

Magnitude of a complex Fourier Coefficient

Normalized value of D

Direct current

Frequency

General Dynamics Corporation

Current magnitude

Current in the frequency domain

Imaginary component of A

(.I)°'s

Jacobian matrix

i,j element of the Jacobian matrix

Inductance

K

K

6)

P

Q

R

Re[A]

S

SCR

O

THD

f

e

v

v

x

z

iv

Substitiute for {[ 4/(LC) - (1/RC) 2 ]°'s}/2

Magnitude of a coefficient of a partial fraction

Complex coefficient of a partial fraction

Imaginary component of a complex frequency

Poles of a circuit

Real power

Reactive volt-ampere

Resistance

Real component of A

Complex frequency

Silicon controlled rectifier

Substitute for 1/(2RC)

Total harmonic distortion

Time constant

Period of a sinusodial waveform

Angle of a complex Fourier coefficient

Angle of a coefficient of a partial fraction

Voltage magnitude

Voltage in the frequency domain

Reactance

Complex impedance

Development of a Single-Phase HarmonicPower Flow Program to Study the 20 kHz

A_ Power System for Large Spacecraft

L. Alan Kraft*

Valparaiso University

Valparaiso, Indiana 46383

Abstract

and M. David Kankam

National Aeronautics and

Space AdministrationLewis Research Center

Cleveland, Ohio 44135

This report describes the development of a software to aid in the design and analysis of AC

power systems for large spacecraft. The algorithm is an improved version of Electric Power

Research Institute's (EPRI) harmonic power flow program, "HARMFLD", used for the study of

AC power quality. The new program is applicable to three-phase systems typified by terrestrial

power systems, and single-phase systems characteristic of space power systems. The modified

"HARMFI.D" accommodates system operating frequencies ranging from terrestrial 60 hz to and

beyond aerospace 20 kHz, and can handle both source and load-end harmonic distortions.

Comparison of simulation and test results of a representative spacecraft power system shows

a satisfactory correlation. Recommendations are made for the direction of future improvements to

the software, to enhance its usefulness to power system designers and analysts.

The advent of large spacecraft has resulted in their increased electrical energy consumption.

Additionally, planned space exploration will require more capability for extended human and

equipment support. These requirements are well served by the AC power system. However, the

presence of switching devices in the converters at the source end of currently proposed, spacecraft

AC system and their connected nonlinear loads can cause several problems in the power system.

One of the more serious problems is the occurrence of harmonic resonance. This is a phenomenon

whereby the harmonics generated by the switching devices and the nonlinear loads excite resonant

modes caused by cancellation or near cancellation of inductive reactance by capacitive reactance.

• 1990 NASA/ASEE Summer Faculty Fellow at Lewis Research Center.

Among theproblems caused by voltageand currentresonance inapower system areinsulation

failuredue toovcrvoltagc,equipment malfunctionresultingfrom high frequencyand multiplezero-

crossingsof distortedwaveforms, radio-frequencynoise,and transmissionand equipment current

overloads. Generally,thegeneratedharmonics tendtoworsen theoverallpower quality.

This report describes a harmonic power flow prOgram, "HARMFLO," for identifying and

evaluating resonance problems, among others, which can occur in either terrestrial or large

spacecraft power systems. The program accepts both three-phase systems commonly found in

terrestrial power systems, and single-phase systems characteristic of spacecraft power systems.

The applicability of the program encompasses frequencies ranging from terrestrial 60 Hz to

aerospace 20 kHz and beyond, and harmonic distortion at the source - and load-ends of the power

system.

In its present phase, the harmonic power flow program contains a mathematical model of a

Mapham inverter [1], a key subsystem of a representative 20 kHz spacecraft power system used in

developing and testing the algorithm. In consideration of criticality of payload weight, the salient

features of lightweight and simple design, with a minimum number of components, make the

Mapham inverter an attractive source of AC power for spacecraft applications. Inclusion of the

inverter model in the algorithm can expedite required performance analysis of the spacecraft power

systems during their design phase. The design and operation of the Mapham inverter is well

documented in the literature [1-3]. Hence, only the particular aspects of the inverter operation

which impacts its modelling are discussed in this report.

Another essential subsystem of the representative power system used is the single-phase

voltage controlled rectifier. As a potential source of harmonic current with attendant problems in

the power system, the mathematical model for this rectifier is also developed.

Simulated preliminary results obtained from the algorithm compare favorably with published

test results. Recommendations included in the report point to future enhancements to the power

flow program to make it more useful to power system designers and analysts.

2

2. Operation Of the Mapham Inverter

The circuit diagram of a

Mapham inverter is shown in

Figure 1. The SCR's are

switched at the desired output

frequency, f.. This switching

reverses the current through the

capacitor, C r. This current

resonates at a frequency

determined by the series

combination of the inductor, L r , and C r .

The reversing of the current at a time

when the resonating current is in its

negative half cycle will sustain the

inverter's operation as an AC source.

Figures 2 through 4, which can be found

in the publication by Sundberg, Brush,

Button and Patterson [3], show the

reversing currents, ILl and I_, though the

inductors, Lr/2, and the total resulting

current through C r. It is important to note

that unless the resonant frequency, f r, is

the same as f., a distortion occurs in the

capacitor current waveform. This

distortion results in harmonic currents

which are injected into the power system

by the inverter. Since the impedance of

the load placed across the output

terminals of the inverter will change f r ,

loading will affect the level of injected

harmonic current.

E_ L r12

y _ Lr,2 l_j_t_._Csl Lr'2

Vo

Figure 1 - Circuit Diagram of a Mapham Inverter.

_'"/. \

!i.Ja-e6 4u.u era-o= on-e•

5r==8 ,_)

Figure 2 - Current ILl

. O.4

=.

!i0o

l\ /\/ /

Ju-os *a-n es.lm H-os

T1NI (lie•mud=)

Figure 3 - Current I,=.

!.._/ \/

i

-o.o'=s.eo .s.es el-el •=-es

¢ztcl (In*_m_)

Figure 4 - Total Current Through Capacitor C,.

3

3. Modellin_o the Mapham Inverter

To determine the injection currents that will occur for a certain power system

configuration, it is necessary to find fr for the inverter under the desired loading. Under

no-load conditions, the resonant frequency is given by

1fr =

LVZ-d

When the inverter is connected to the power system, the system appears as an equivalent

impedance of Z_ = Req + JXeq connected across the output terminals of the inverter

as shown in Figure 5. In the frequency domain, the input current resulting from a step

input voltage has the form

Elm Sm+ ... + 81S + a 0

b,s" + ... * bls + bo

where, m < n and the coefficients of the two

polynomials depend on whether the system

equivalent is capacitive or inductive (i.e.,

Iz,,,,.=.=l oR c ..

l o L.._

Figure 5 - Power System Equivalent.

whether Z_ = Req - JX_ or Z_ = R,,q + JXeq). The details concerning the derivation

of these polynomials can be found in Appendix A.

If the system equivalent is inductive then n = 5, and the coefficients of the

denominator are:

4

bs = (L_ ÷ L.q)L_C_C, ,

b, = R°qL_C.C, ,

LI + Leg 1b 3 = (L l + L#q) Cs÷ 1 + Lr(c r + C m) ,Lm

R,_L, (co+ c,)b2 = a,qc s +

• Lm

L 1 + L,qb I = I + , and

Lm

However, if the equivalent is capacitive then n = 6, and the coefficients of the

denominator are:

b 6 = L1LrCeqCrC s ,

b s : R.qLzCe_CrC s ,

b 4 = LlC_C ,+LrCzC s+LzceqC r+LzC_c s+ LILrCeq (C r÷c'7)L,.

b3 = ReqCeqC s+ ReqLzCeq (C s+ Cz)Lm

5

=Lm

b, = a._C._ andLm

Once the coefficients are computed, the denominator can then be factored into

(s. + P.) (s._l + P.-1) ... (sz + P2) (el + P1) •

The roots of this polynomial (i.e., the P's) represent the poles of the network. Since the

network is primarily a series L-C circuit, at least two of the poles will form a complex pair

with the form:

(s. + a. + j_.) (s.__ + a s- je.) •

Since n > m, this polynomial is said to be a proper rational polynomial and can be

expressed in partial fraction form:

Z. £." KIKn +...+ + ÷...+

"sn+P. sz+az-j(_ z sz+az+J _), SI+PI

6

The function can then be returned to the time domain by using inverse Laplace

transforms. The inverse transform for the complex pair is

2-I[ s+=-j_KLO + s+=+j_KL-e ] = 2Ke-'tcos (_t+8)(1)

This yields a time domain response for the current through C r which is

responses due

Ic, (t) = loe-"'tsin (_zt + e z) + to the otherpol es

Once the natural resonate frequency

and its corresponding period

Jr

1 2_

fr _ Jr

0.I-

0.4

o

M-0.4

-0.10

RBGIONS POR MODR_

I ' I z (c:)i r L --

J!\ // k\ I (L2)

SE-05

TZME (SecoMs)

are known, the components of the Figure6- Capacitor Current, C,; Inductor Currents, I,,

generated harmonic currents can be and IL2 with Indicated Regions for Modelling.

determined. Figure 6 shows one period of the current through C r. Also shown in Figure

6 are the corresponding currents ILl and IL2 which flow through each set of switched

inductors. From this information, the capacitor current can be written as a function of

time over five individual regions of the waveform. The resulting equations are:

7

IC r

-a(_+'%-) sin (_ •= -e

(2)

IC I

(3)

"_ Ct(=) = e

foI --" _ t < _ s 22

(41

IC r(t) -- e

t- \ ,, fj-i Ic-'¢ ,* si_ _ z

-eand

for • j 2

(s)

= --e

for 2

(6)

8

Since the current is now mathematically represented as a continuous, repetitive time

function, it can be represented by a Fourier series. The Fourier series of a repetitive time

function is given by

el

f(t) = Co + ]E_ c-c°sn_ot + dnsinn(_otn-1

where,

tl+T 0

Co _ 1 / f(t) dt,TO C=

(7)

O n

2

T o

CI÷IC 0

/Ca

f(t) cos (no or) dt , (e)

and

d n

2

T o

C 1 ÷ T 0

ICa

f(t) sin (n_ot ) dt . (9)

The Fourier coefficients, c, and d,, are found by integrating Equations (2) through (6)

over one period of the waveform. It should be noted that the magnitude of the current

is assumed to be 1.0 at this point. Once the coefficients are computed, they are then

normalized by

9

D_ _ n81= (Io)

The coefficients are normalized because the magnitude of the current will not be known

until a fundamental power flow is found. The required values of D 1 and 01 are then

known.

To apply these coefficients to the harmonic power flow algorithm, it is first

necessary to understand the formulation of the harmonic power flow algorithm itself. The

details of how this will be done are covered in Section 7 of this report.

10

4. Operation of the Single Phase, Voltage Controlled Rectifier

,

voltage, V x,

shown in Figure

8, it can be

controlled by

changing the

The circuit diagram of a single phase, voltage controlled rectifier is shown in Figure

Since the D.C. output voltage of the rectifier is the average of the "ripple" output

Ys+O

- O--

L

m

Figure 7 - Single Phase, Voltage Controlled Rectifier Circuit.firing angle, e.

This means that,

by adjusting a, the average value of the "ripple" or the D.C. output voltage can be

controlled within certain limits. The SCR's are triggered with a signal which is set by a.

The value of a is determined by the difference between the desired D.C. output voltage

and the actual D.C. output voltage. Once either of the SCR's is triggered, it will conduct

until the voltage on the anode is less than the voltage on the cathode. This occurs when

the input voltage, V _n,drops below the output voltage, Vout, at #, the commutation angle.

Voltage

:1

J_:t

/

....!

i

i

i :l:1

;I

i

/

1

I

i I:' i

t :

: i

j = I :

,. I I:: I

;, I iI I

Time

Vin

Vx

Firing

Angles

Commutation

Angles

Figure 8 - Input Voltage to a Single Phase, Voltage Controlled Rectifier Circuit.

11

5. Modelling the Single Phase, VoIta ae Controlled Rectifier

When either of the SCR's is triggered, the voltage which is applied to the input

terminals of the rectifier is given by

Vz=(t) = Vmsin((_=t+¢) u(t) .

This function becomes the following in the frequency domain:

v,. (s) = v=[ ,,.i,=.,,.¢o.,,].,,=.=; (11)

The entire single phase rectifier circuit in the frequency domain is shown in Figure 9. The

current source, CV(0÷), is needed to account for any initial voltage that might be present

on the capacitor when the SCR is fired at oz. Since I=n(s) is given by

x,.(s)vi° (s) - v=(s)

sT.,(12)

V x(s) must be determined. This is done by writing Kirchhoff's current law at the node

with the unknown voltage, V x(s). L

¥ In(s) &'V(O *) 1-- • Rs¢

Figure 9 - Single Phase, Voltage Controlled Rectifier Circuitin the Frequency Domain.

12

The result is

Vm (ssin= + _acos=)- v,, (s)

v=(s) (s = * _,=,)sCVz(s) + R = st. + CV(O')

i+ 1)v,,(s) sc+ -_ s-_IF= (ssin= +(_=cos_¢) +sLC(s=+w= =) V(O*)

sL ( s= ÷ o)_}(13)

v=, (s)-LC[V,, ( ssin,, + _,cos=) + sLC( s = + ¢_s =) V(O') ]

Ig

" =;)

If Equations (11) and (13) are substituted into Equation (12), the result is

I_=(s)

Vm (ssin= + _,cos=)

( s= * ,_2)

L_[V" (ssin= + _,cos¢) + sLC(S' + ¢_==) V(O °) ]

s =÷_Is . l___l(s =*==')RC LC /

sL

-Rc)[ v= ( ssin=. ,,,,cos= ) ] - s ( s = * _,=,) v(0")

s + i )(sZ.e=)SL S2 * R---C L--_

Vm(s.L -R--'_C)[ssin= + _.cOs=]- (SZ+_)V(0")

s . .-E_s . ___1) ( s = . ,,,=,)RC LC

13

This can now be expressed in partial fraction form:

z,. (s) = K1 z e1 +

2RC 2

K 1/. -e:

2RC 2

+ K2Ze2 K2Z-e 2+

s-jo_, s+jo_.

where,

= IS+ 12RC 2s)

,. I2RC 2

and

K2Ze2 = [s-jo,,]z,..(s) l,,._." .

To simplify the notation, let

o1

2RCand

14

then

K_Le, = [s+e-jA] X,.(s)l,..,.j,

•, (_)(-a+jA+2a)[(-a+jA)sin"+_'c°s"] - [(-a+jA)z+_m=] V(O+)

(j2;.) [(-o+jA)2+_o =]

: (-_)(o+JA)[(,_.cosa-ostn,,) +jAstn2a] - [(o'.A'+_)-J2o).] V(O')

(j2_.) [(o'*;.'*,,)o 2) -J2o;.]

= (--VL=)(a+JA)[((°*c°s=-asina)+jAsin=]

(j2k) [ (o2+A2*_# 2) -j2oA]

V(O °)j2A

and

K_L% = [s-j_),] x_=(s) l,.:,,

)(j(o,÷ _c) (j_),sin= ÷(a.cos=) - (-(0.2+(_.2) v(o')

( .°._)-0,2+3-_ + (j20.)

( vo/l_ +_..)(_.,.)- (o)v(o.)L /% RC

°'1- _;) ÷3-E_ (j2(_,)

-_ l+j_.) COS (a--_) +jsln (a--_)

15

Once the complex coefficients, K 1 and K 2, are computed, the transformation back into

the time domain is found using Equation (1). Now that the input current is knownin the

time domain, Equations (7), (8), and (9) can be used to find the Fourier coefficients of the

current. It is important to note that, because the total current is known at this point,

normalization of the rectifier coefficients is not needed.

These coefficients, like those found for the Mapham inverter, must be incorporated

into the harmonic power flow program. To accomplish these goals, the following is a brief

summary of the development of the harmonic power flow program.

16

6. The Harmonic Power Flow Algorithm

The harmonic power flow algorithm used in this work is the one developed by

Heydt and Xia [4,5] and later modified by Grady [6]. The algorithm uses a standard

Newton-Raphson formulation for a power flow program. The basic algorithm is:

1. Select an initial solution vector for all bus voltages, [V (o}], where

element j is I Vjl and e j or the complex voltage at bus J.

2. Using the voltage vector, calculate

A Pj = pjBcheduled _pjca]cu]ated

A Qj = QjSChedule__Qfalc.la_,d .

If A P and A Q for all busses are < _, a specified tolerance, a solution

has been reached; and, the iterative process is stopped; and, the

output is printed.

3. Compute the elements of the Jacobian matrix which are defined by

[J_]

8P i 8P i ]

= av,80i 80i __

17

4. Using [A p, _, Q] T = [j ] [A V, A 6 ] T, calculate A V and & 6.

5. Update the solution voltage vector,

= + [Av, Ae] t

6. Repeat Steps 2 through 5.

This basic power flow algorithm was modified by Heydt, Xia and Grady in the following

manner.

1. The basic algorithm described above is run on the system for the

fundamental frequency only.

, The solution voltage vector obtained in Step 1 is used as the initial

value in the harmonic power flow. Therefore, the initial voltage

vector is

V(1)

V(2)

V(a)

IF(,-)

where, V (i) represents the bus voltage for the i m harmonic and J

are the firing and commutation angles for the converters.

18

3. Using the solution vector, calculate

APj = p/ch. u1.,_ pI.1o.l.t.

/% I (i) = I (I} scheduled_ I {I) calculated (14)

where, I(i) scheduled is the scheduled ith harmonic current at each bus,

and I °) ca,cu,a,edis the calculated i th harmonic current at each bus.

It should be noted that the scheduled current for all busses except

harmonic source (ie., nonlinear devices) is 0. If ,%P, ,%Q, and ,%I are

< E, a specified tolerance, for all busses, the solution has been

reached; and, the algorithm is finished.

4. Calculate the elements of the modified Jacobian matrix. The

Jacobian matrix is modified by formatting it as follows

[J]

j(1)

TG(2,1)

TG (h,1)

TG(1,1)

j(2) . . . j(n) 0

TG(2,2) . . . TG(2,h) H(2)

TG(h,2) . . . TG(h,h) H(h)

TG(_,_) . . . TG(_,h) H(1)

19

where,

j/j (m) .a vj (m)

a oj (rn)

a vj (m)

vja6j

a oi (m)

and

TGij(m,n)

_(m)

J

a (lia_ (")

n

aR,,(li (m))a=j

a_n,(_j(m))o_aj

an,,pj _")_a aj (n)

a 6j (n)

aRe (Ii (m))

a Im(Ii (m))

a_j

, Using [A P, A Q, A I] T = [j] [,_ V, A 6, A or,A 13]T, calculate & V, A 8, A 0¢,

6. Update

[V (i}, 6 {i}, (7.{i}, p {i}] T = [V {i-1}, 6 {i-1}, Or,{i-1}, p {i-1}]T + [4 V, A 6, A (x, & p] T

7. Repeat Steps 3through 6.

The modifications introduced in the basic power flow make the HARMFLO different fromothers which use the fundamental power and harmonic current responses of thenonlinear devices to solve the voltage levels within a power system [7]. Furthermore,the HARMFLO differs from the Alternate Transients Program (ATP) version of the

Electromagnetic Transient Program (EMTP) [8]. The ATP version is used in analysis inwhich the nonlinear loads are represented by harmonic current injections at desired

nodes within the power system.20

7. Modi_ing the Harmonic Power Flow Algorithm

Since three phase, synchronous machines used in terrestrial power systems are

designed to produce an undistorted 60 Hz sine wave under normal operating conditions,

these electric power producing machines do not inject harmonic currents into the power

grid. Because of this fact, the harmonic power flow program described above does not

include the ability of the electric power sources to inject harmonic currents into the

electrical system. Inverters, particularly the Mapham inverter, are, however, large

producers of harmonic injection currents. These inverters inject the harmonic current at

the bus where they are connected to the system. This change in sources required a

modification to the harmonic power flow program.

As previously stated, the Fourier coefficients of the capacitor current, I_, are

computed by inserting Equations (2) through (6) into Equations (7) through (9) and

integrating over the appropriate ranges. The resulting complex coefficients are then

normalized by Equation (10). This process is summarized by

= Re (Dj (1) Lej (1) e j_)°c + Dj (2) Lej (2) e j2_°t + ...)

[ DJ(') I (1)) eJ2_ocIj ( t )"°r'_lized = Re (1 Z 0° e j°'°t + _ Dj(----TIZ(8j(2) _ nOj + ...)

= Re (1ZO ° e j(*°c + Di(2)lZe21e j2'_ot ÷ ...) .

Therefore, the magnitudes of the actual injection currents at every inverter bus for each

harmonic can be determined by multiplying the magnitude of the fundamental source

current at the bus by the appropriate Fourier coefficient. Thus, the scheduled injection

current for an inverter bus j during the solution of the harmonic power flow can be

21

determined as follows by using the fundamental current magnitude and angle, I (_) and

l(1), which is found in Step 3 of the modified algorithm described in Section 6:

.z'.l( t ) ,ch_,l_ = (i(z) Z$._(_) ) (Ij ( t ),,o,==zl=_)

= (Ij (1)z_j (z)) [Re(1/O ° ejOoC + D:I(=)/Le:I(=)/e :12"=_ ÷ ....

= Re[i:l(1)/_j(z)e j'ot + I_ (z) (D./(=)') Z (0./(=)1 +nlll./(=)) e j

= .Re [ij(z)ZSj(1) eJ,.,ot + ij(=),,,$j(") eJ='o =: . ...] .

The injected harmonic currents are then placed in the appropriate locations in Equation

(14). In order to place the harmonic injection currents into Equation (14), it is necessary

to represent them in rectangular form (i.e., X + jY). Therefore, the current at each

harmonic is found by

Ij(m) scheduled (m) (m)).ee = Ij cos (_j

(z_(') ) (D: (=)1) cos(e_(,,)'- nej (*))

( Ij

and, in a similar manner,

[11,.) ]r (m) scheduled (1) (m)/2 (m)/2 _, ¢1 (m"j,z= = ( Ij ) ( _ Cj - dj ) sin can- I--_,_,i -n01(i) .

22

The harmonic power flow program does model nonlinear devices such as rectifiers.

The problem is that these rectifiers are three phase. Three phase rectifiers do not

mathematically function like a single phase rectifier. For this reason, modifications in the

harmonic power flow code are necessary. The modifications needed to successfully

incorporate single phase rectifiers into the harmonic power flow code are simply to place

the various harmonic injection currents described in Section 5 of this report into the

scheduled current vector:

I (m) scheduled r (m) schodUlmdJ,R, = Cm and _ j,zm = dm•

Once this is done for all Mapham inverters and single phase rectifiers, the

harmonic power flow program is allowed to proceed per the algorithm described in

Section 4.

23

8. Examples

To demonstrate the practicality of the theories presented in this report, the

following examples are offered. The first example will investigate the voltage distortion

experienced in the power system depicted in Figure 10. This example is drawn from data

obtained from the

General Dynamics

Corporation (GDC) 20

kHz Testbed which was

developed to study the

proposed 20 kHz, A.C.

power system. The

Tronsm£ss£on Line (50 mtere lonl.)

._.L -L

Figure 10 - General Dynamics Corporation (GDC) 20 kHz Testbed PowerSystem.

inverter is a Mapham inverter set which will produce 440 Vr_

have the following electrical characteristics:

at 20 kHz. This inverter will

L z = 16 I_H ,

c_ = 1.71 l_Y,

C_ = 2.0 pF ,

L I = 1.8 _H,

Lm = 1.0 m H , and

f8 = 20 kHz .

The transmission line is a 50 meter length of stripline designed cable which has the

following electrical characteristics:

24

Resistance = 1.043 at_meter

Inductance = 0.027meter

Capacitance = 0.003 --El--meter

• and

The load is standard A.C. load which will be varied from 2 kVA to 10 kVA. The load will

maintain a constant power factor of 80% lagging throughout this example. Figure 11

shows the data formatted for the harmonic power flow program. The load was varied

from S = 0.16 + j 0.12 p.u. to S = 0.80 + j 0.60 p.u. The resulting Total Harmonic

Distortion (THD) of the bus voltage, defined as

v_+_ = _ (vi(n))2n-2 Vi (I) '

was used to

generate the

graph shown in

Figure 12. Note

that the results of

tests conducted

by Sundberg, et.

al. are also

1

NASA Test System2

Test System for Paper #899383 of the IECEC '89 Conference, Vol. I3

1 Sourcl 0816.0e-6 1.71e-6 2.0e-6 1.8e-6

3 Loadl 039999

4

1 3 0.54 1.75

1 0 0.0 -5305.

3 0 0.0 -5305.

3 0 266.7 200.01 0 0.0 649°09

99995

g 590 080 I 000 015

O.Oe-O 2.0e+424.0 18.0

100.00 0.0440.0 1.Oe+4

Figure 11 - Input Data from the GDC 20 kHz Testbed for the Harmonic Powerpresented on the Flow.

graph in Figure

12. Examination of the graph in Figure 12 reveals that the harmonic power flow does

yield results which are very close to the values obtained experimentally. It should also

be noted that the results obtained from the harmonic power flow program are

exceptionally sensitive to the impedance values. It was found that very small changes in

the data will cause relatively large changes in the output.

25

The second example will also use the system described in Figure 10. This time the

switching frequency will be varied to get the ratio fr/f= tO vary from 0.6 to 0.95. The data

for this example remains the same as that found in Figure 11 with the exception of the

switching frequency. This value is varied as described above; and, the resulting THD of

the bus voltage is compared to that measure by Sundberg, et. al. in Figure 13.

p.f. - 0.8, lagging

10 8undl>ezg

e¢. i1.

---" 8 --e--0_0 _ ) t_zmonJ.c powezv

6 _' Flow (Vthd)..... _.'....... _ ............ ....(..÷...

4 ,.' _ m,,=monic Po',,e=P'low (l chd)

0 _'_ "-"e--°0 2 4 6 8 10

KVA

"myerS, e: uses • 2.0 uF 8eztel C•p•citoz

Figure 12 - Comparison of Results of Bus Voltage THD versus KVA Loading fromHarmonic Power Flow Program and Test Data from the GDC 20 kHz Testbed.

Current THD versus KVA Loading is also Included.

p f - 08, lagging

16 Sundberg

14 .,!........_ eC. -1.

v i0 _ ", '""" "_ T1ow (vcbd)_'. .,-'" \ .... ,,.-+...

e _ t -

;f

6 _z

2 _

006 07 08 09

fsn

Invozcex uloe m 2.0 uF Soziom C•p•citoz

Figure 13 - Comparison of Results of Bus Voltage THD versus f,. from Harmonic

Power Flow Program and Test Data from the GDC 20 kHz Testbed.

26

9. Conclusions and Recommendations

This report for the 1990 Summer Faculty Fellowship Program documents the

development of software which will benefit the design and analysis of large A.C.

spacecraft power systems. Early results indicate that the models developed are quite

adequate for the intended goals. This is demonstrated by the acceptable correlation of

the results exhibited in Section 8. Although the results are not exact, the discrepancies

are relatively small. Given the small discrepancies and the limited test results from the

GDC 20 kHz Testbed for comparison, further comparisons are necessary before accurate

conclusions concerning program accuracy can be drawn. Once the GDC 20 kHz Testbed

is made operational in its new location, an organized set of tests should be devised which

will thoroughly evaluate the models developed in this report. This evaluation will help

refine the models.

As a result of the work cited in this report, the following areas are recommended

for further investigation and development:

• complete voltage control for voltage controlled rectifier,

• model series and parallel operation of Mapham inverters, and

• model bidirectional receivers.

Implementation of the voltage controlled rectifier model discussed in Sections 4 and

5 was started during this time frame. The model, however, was not completed due to

time constraints. The model, which is presently incorporated into the harmonic power

flow program, is a single phase, full wave rectifier. The model for this device is exactly

that of the single phase, voltage controlled rectifier with the exception that the voltage is

determined by the system configuration and not controlled to a desired level. The work

required to implement voltage control appears to be straightforward.

27

Mapham inverters are operated in series to increase the total output voltage of the

source. The modelling of this operating configuration must be investigated so that this

mode of operation can be included in future design reviews.

The bidirectional receiver unit is a device which is proposed for use on space

vehicles. Modelling this device in the harmonic power flow program is important for a

complete analysis of any proposed power system. For this reason, it is recommended

that work be conducted in this area as well as those mentioned above.

28

LIST OF REFERENCES

[1] N. Mapham, "An SCR Inverter with Good Regulation and Sine-Wave Output,"

IEEE Trans. on Industry and General Applications, IGA-3, no. 2, pp. 176-187,

March/April 1967.

[2] A. S. Brush, R. C. Sundberg, R. M. Button, "Frequency Domain Model for

Analysis of Paralleled, Series-Output-Connected Mapham Inverters," 24 th

Intersociety Energy Conversion Engineering Conference, Washington, D.C.,

August 1989.

[3] R. C. Sundberg, A. S. Brush, R. M. Button, A. G. Patterson, "Distortion and

Regulation Characterization of a Mapham Inverter," 24 th Intersociety Energy

Conversion Engineering Conference, Washington, D.C., August 1989.

[4] D. Xia, G. T. Heydt, "Harmonic Power Flow Studies Part 1 - Formulation and

Solution," IEEE Trans. on Power Apparatus and Systems, vol. PAS-101, no. 6,

pp. 1257-65, June 1982.

[5] D. Xia, G. T. Heydt, "Harmonic Power Flow Studies Part II - Implementation and

Practical Applications," IEEE Trans. on Power Apparatus and Systems, vol. PAS-

101, no. 6, pp. 1266-70, June 1982.

[6] W. M. Grady, "Harmonic Power Flow Studies," Ph. D. Thesis, Purdue University,

West LaFayette, IN, August 1983.

[7] F. Williamson, G. B. Sheble, "Harmonic Analysis of Spacecraft Power Systems

Using a Personal Computer," 24 th IECEC, 1989.

[8] R. Leskovich, I. G. Hansen, "The Effects of Nonlinear Loading Upon the Space

Station Freedom 20 kHz Power System, "24 th IECEC, August, 1989.

29

Appendix A

Derivation of Coefficients of System Responses

To find the polynomials of the input current it is simply a matter of employing

Ohm's Law in the frequency domain,

zc (s) =v_= (s)

When the Mapham inverter is connected to a power system which can be modelled as

a series R-L circuit, the equivalent input impedance of the entire network is the impedance

Zoq shown in Figure A.I. The following steps are used to find Z,q which is connected to

the source.

sL , ZlsC $ sL I

V in (s) s s I Z e.(s) J

Figure A.1 - Total System Connected to the Source.

1. Series of sLeq, SLeq, and Req yields Z l(s ).

Z 1 ( S ) = Req + S ( L 1 + LW) .

30

2. Parallel of Zl(s ) and sL m yields Z2(s ).

za(s)z_ (s) (sr..)

z1(s) + sL.

s2 L_ (LI + L_) + sR,qL m

s (L I + L,_ + L m) + a_

3. Series of Z 2(s) and 1 /sC, yields Z 3(s).

z, (s) - 1 + z,(s)S C s

1 S2Lm (LI + Leq) + SReqL m-- +

sC s s (L l + Leq+ L m) + Req

s scsL m(L 1 +Leq) +s2R,qLmCs+S(L l+L,q

s2Cs (L I + L,q+ L m) + SR._C a

+ L m ) + R_I

4. Parallel of Z3(s ) and 1 /C, yields Z4(s ).

z4(s)(i)= z,Cs)

1+ z,(s)

sc,

S3CsL,. (L I + Leq) + S2ReqLmCs + S (11 ÷ Leq+ L m) + Req

s4CsCzLm(L 1 +Leq) + S3ReqLmCzCs+ s2Cs(L l + Leq+ L m) + SReq(C z +c,)

81

5. Series Z 4 (s) and sL, yields Z eq (s).

z,,v(,s) - ez,_ .,. z4 (0)

sz mSLtLmCmCz(LI+L'q ) *B4LtLmCeCrlt'q+Js[LmCm{LJ+L'q) '+L_r(C'* Ct) (LI+L_+Lm)| _'OS(It'NIL#(C°eCz) +LmCa]J+m(L/'+L_4'Lm) "&"_

B' z.aC. Cz (Z.j + Lev) • #*L,C, Ca_w" w# [ (Cr ÷ C,) (Z., *Z.w* Lm) ] * maw ( C, + C,)

Since the Laplace transform of the switched input voltage is

] - 1 ,s

then the input current is

3.

it(s) _ sZ_

- --(s)

#3 L.C.Cz ( £j + L,_) * s2LmC, Cta,q + s [ ( C z + C.) ( Lj + L,q + L m) ] + a,q ( C. + Cs)

#SLzLmC#C_r (LI "+L,,_) ", mILzLmC,,CzRev * m_ [LBCm (LI " L._) _' Lt ( C, "."C r) (/'.1 _"Le.v * Lm) ] _' #Z('qW [Lz ( C'o" C-'r) " LmCm] |'" a(£l "Lw÷ Lm} " i'oq'

The denominator of the above equation yields the coefficients which are necessary to find

the natural response of the input current.

When the power system equivalent is a series R-C circuit rather than the series

R-L which is shown above, the method outlined above is used to find the coefficients of

the polynomial in the denominator of the current's function in the frequency domain.

82

Form ApprovedREPORT DOCUMENTATION PAGE OMB No. 0704-0188

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

November 1991 Technical Memorandum

4. TITLE AND SUBTITLE S. FUNDING NUMBERS

Development of a Single-Phase Harmonic Power Flow Program to Study the

20 kHz A.C. Power System for Large Spacecraft

S. AUTHOR(S)

L. Alan Kraft and M. David Kankam

7, PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAMES(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, D.C. 20546-0001

WU-506-41-41

8. PERFORMING ORGANIZATIONREPORT NUMBER

E-6687

10. SPONSORING/MONITORINGAGENCY REPORT NUMBER

NASA TM-105326

11. SUPPLEMENTARY NOTES

L. Alan Kraft, Valparaiso University, Valparaiso, Indiana 46383 and NASA/ASEE Summer Faculty Fellow at Lewis

Research Center in 1990. M. David Kankam, NASA Lewis Research Center. Responsible person, M. David Kankam,

(216) 433-6143.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category 20

12b. DISTRIBUTION CODE

13. ABSTRACT (Max/mum 200 words)

This report describes the development of a software to aid in the design and analysis of AC power systems for largespacecraft. The algorithm is an improved version of Electric Power Research Institute's (EPRI) harmonic power flowprogram, "HARMFLO", used for the study of AC power quality. The new program is applicable to three-phasesystems typified by terrestrial power systems, and single-phase systems characteristic of space power systems. Themodified "HARMFLO" accommodates system operating frequencies ranging from terrestrial 60 hz to and beyondaerospace 20 kl-Iz, and can handle both source and load-end harmonic distortions. Comparison of simulation and testresults of a representative spacecraft power system shows a satisfactory correlation. Recommendations are made forthe direction of future improvements to the software, to enhance its usefulness to power system designers and analysts.

14. SUBJECT TERMS

Space power systems; Harmonic power flow; Power systems analysis

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